17. Let x1, x2, x3 be the roots of the equation x + 3x + 5 = 0. What is the value of the expression
(x1+1/x1)(x2+1/x2)(x3+1/x3)?
Answers
Given: x1, x2, x3 be the roots of the equation x³ + 3x + 5 = 0.
To find: Value of the expression (x1+1/x1)(x2+1/x2)(x3+1/x3)
Solution:
- As we have given the equation, x³ + 3x + 5 = 0 and roots as x1, x2, x3, so:
x1 + x2 + x3 = -b/a = 0
x1x2 + x2x3 + x1x3 = c/a = 3
x1x2x3 = -d/a = -5
- So now, we have to find value of:
(x1+1/x1) (x2+1/x2) (x3+1/x3)
- Simplifying it, we get:
((x1²+1)/x1) x ((x2²+1)/x2) x ((x3²+1)/x3)
(x1²+1)(x2²+1)(x3²+1) / x1x2x3
(x1²x2² + x1² + x2² + 1) (x3²+1) / x1x2x3
(x1²x2²x3² + x1²x2² + x1²x3² + x1² + x2²x3² + x2² + x3² + 1) / x1x2x3
- Now we know the formula of (a+b+c)² = (a²+b²+c² +2ab+2bc+2ca)
- So we can write (x1²x2² + x1²x3² + x2²x3²) as
(x1²x2²x3² + x1²+ x2² + x3² + 1 +{ (x1x2 +x2x3 + x1x3)² - 2(x1x2x2x3 + x1x3x3x2 + x2x3x1x1 } ) / x1x2x3
- Now putting the values on this, we get:
(-5)² + x1²+ x2² + x3² + 1 + { (3)² - 2((-5)x2 + (-5)x3 + (-5)x1 } ) / -5
25 + x1²+ x2² + x3² + 1 + (9 + 10x2 + 10x3 + 10x1) / -5
25 + x1²+ x2² + x3² + 1 + 9 + 10x2 + 10x3 + 10x1 / -5
35 + x1²+ x2² + x3² + 10(x1 + x2 + x3) / -5
- Now again using the formula of (a+b+c)² = (a²+b²+c² +2ab+2bc+2ca)
35 + (x1+x2+x3)² - 2(x1x2+x2x3+x1x3) + 10(x1 + x2 + x3) / -5
35 + 0 - 2(3) + 10(0) / -5
35 - 6 / -5
29 / -5
-29/5
Answer:
So the value of the expression (x1+1/x1)(x2+1/x2)(x3+1/x3) is -29/5
Answer:
Step-by-step explanation:
Hope it helps u!....
